Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme

WAVELET HMTFOR NOISEREDUCTION

http://dsp.vscht.cz/

ICT Prague

Introduction

Wavelet ShrinkageWS Scheme

NormalShrink

Noise Reductionvia HMTsPersistence & Clustering

Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography

ICTC Prague 2008 1 / 24

WAVELET-BASEDHIDDEN MARKOV TREES

FOR IMAGE NOISE REDUCTION

Eva Hošt’álková & Aleš Procházka

Institute of Chemical Technology, PragueDept of Computing and Control Engineering

Technical Computing Prague 2008



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction

2 Wavelet Shrinkage (WS)Wavelet Shrinkage SchemeNormalShrink Method

3 Noise Reduction via Hidden Markov Trees (HMTs)Persistence and Clustering PropertiesWavelet-Based HMT ModelsNoise Reduction

4 Results

5 Conclusions

6 Selected Bibliography



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Introduction

Applications of Noise ReductionImage enhancementPreprocessing step to other techniques(e.g. segmentation, edge detection)

Noise Reduction via Wavelet Shrinkage

+ Recovering signals from additive Gaussian noise- Thresholding w. coefficients without considering their

dependencies⇒ artifacts, blurred edges

Noise Reduction via Wavelet-Based HMTsHidden Markov Tree (HMT) modelsAiming to capture these dependencies



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Introduction








ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Introduction








ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Introduction

Other Applications of Wavelet-Based HMTsImage segmentation (texture features)Edge detection, signal prediction etc.

Image Data

(a) MANDRILL IMAGE (b) CUT OUT (c) NOISY IMAGE

The mandrill image, a 240×240 cut out, corruption byiid Gaussian noise



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Introduction

Other Applications of Wavelet-Based HMTsImage segmentation (texture features)Edge detection, signal prediction etc.

Image Data

(a) MANDRILL IMAGE (b) CUT OUT (c) NOISY IMAGE

The mandrill image, a 240×240 cut out, corruption byiid Gaussian noise



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Wavelet Shrinkage Scheme

Wavelet Shrinkage Algorithm

NoisyImage

NoisyImage

WaveletAnalysisWaveletAnalysisWaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

The NormalShrink Method

WaveletSynthesisWavelet

SynthesisWavelet

Synthesis

Image

LoLo2

LoLo1LoHi1HiLo1HiHi1

LoHi2HiLo2HiHi2

The Haar wavelet transformThe Haar wavelet transform

DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImage

NoisyImage

WaveletAnalysis

WaveletAnalysisWaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients



SynthesisWavelet

Synthesis

Image

LoLo2


LoHi2HiLo2HiHi2


DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImage

NoisyImage

WaveletAnalysis


Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients



SynthesisWavelet

Synthesis

Image

LoLo2


LoHi2HiLo2HiHi2


DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImage

NoisyImage

WaveletAnalysis


Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


WaveletSynthesis


Synthesis

Image

LoLo2


LoHi2HiLo2HiHi2


DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImageNoisyImage

WaveletAnalysis


Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


WaveletSynthesis


Synthesis

Image

LoLo2


LoHi2HiLo2HiHi2


DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImage

NoisyImage

WaveletAnalysis

WaveletAnalysis

WaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


WaveletSynthesis

WaveletSynthesis

WaveletSynthesis

Image

LoLo2


LoHi2HiLo2HiHi2

The Haar wavelet transform


DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography




NoisyImage

NoisyImage


WaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients



Synthesis

WaveletSynthesis

Image

LoLo2


LoHi2HiLo2HiHi2



DenoisedImage

DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


NormalShrink Method

NormalShrinkSubband-adaptive threshold computation

MAD estimate of iid Gaussian noise std. deviation

σ̂nmad = median{|whh1

1 |, |whh12 |, . . . , |whh1

N/4|}/0.6745

whh1 . . . HiHi wavelet coefficient of level 1 (noisedominated), N . . . image sizeRobust against large deviations of noise variance

Soft shrinkage function

SOFT THRESHOLDING

δ(s)

−δ(s)

−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)

−2 δ(s)

−δ(s)

0

δ(s)

2 δ(s)

3 δ(s)

before thr.after thr.



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


NormalShrink Method




1 |, |whh12 |, . . . , |whh1

N/4|}/0.6745


Soft shrinkage function

SOFT THRESHOLDING

δ(s)

−δ(s)

−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)

−2 δ(s)

−δ(s)

0

δ(s)

2 δ(s)

3 δ(s)




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


NormalShrink Method




1 |, |whh12 |, . . . , |whh1

N/4|}/0.6745


Soft shrinkage functionSOFT THRESHOLDING

δ(s)

−δ(s)

−3 δ(s) −2 δ(s) −δ(s) 0 δ(s) 2 δ(s) 3 δ(s)−3 δ(s)

−2 δ(s)

−δ(s)

0

δ(s)

2 δ(s)

3 δ(s)




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Persistence and Clustering Properties

Interdependencies of the DWT CoefficientsShrinkage methods assume the DWT to de-correlatesignals thoroughly (incorrect)DWT coefficients reveal clustering and persistence

Persistence & Clustering Properties

Clustering within scaleWe expect large (small) coefficients in the vicinity ofa large (small) coef.

Persistence across scaleParent-child relationsWe expect a large (small) parent coef. to have large(small) children

Both captured by the HMT models



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



Interdependencies of the DWT CoefficientsShrinkage methods assume the DWT to de-correlatesignals thoroughly (incorrect)DWT coefficients reveal clustering and persistence

Persistence & Clustering Properties

Clustering within scaleWe expect large (small) coefficients in the vicinity ofa large (small) coef.

Persistence across scaleParent-child relationsWe expect a large (small) parent coef. to have large(small) children

Both captured by the HMT models



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



Persistence and Clustering

LL3

LH2

HH2

HL1

HH1LH

1

HL2

2D: each parent coefficient p(i) has four children iHMT connects the hidden states Si , Sp(i) - not theactual coefficients values wi , wp(i)



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Wavelet-Based HMT Models

Histograms of Wavelet Coefficients

−0.5 0 0.50

0.5

1

1.5

2

(a) LH1 COEFFS HISTOGRAM

Histogram State S=1 State S=2 Marginal PDF

−0.5 0 0.50

0.5

1

1.5

2

2.5

(b) HL1 COEFFS HISTOGRAM

−0.5 0 0.50

0.5

1

1.5

2

2.5

(c) HH1 COEFFS HISTOGRAM

Probability Distribution of Wavelet CoefficientsNon-Gaussian distribution (peaky and heavy tailed)M-component mixture of conditional G. distributionsG(µi,m, σ

2i,m) associated with hidden states Si = m

For M = 2 (2-state models) m = 1,2



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



Histograms of Wavelet Coefficients

−0.5 0 0.50

0.5

1

1.5

2

(a) LH1 COEFFS HISTOGRAM

Histogram State S=1 State S=2 Marginal PDF

−0.5 0 0.50

0.5

1

1.5

2

2.5

(b) HL1 COEFFS HISTOGRAM

−0.5 0 0.50

0.5

1

1.5

2

2.5

(c) HH1 COEFFS HISTOGRAM

Probability Distribution of Wavelet CoefficientsNon-Gaussian distribution (peaky and heavy tailed)M-component mixture of conditional G. distributionsG(µi,m, σ

2i,m) associated with hidden states Si = m

For M = 2 (2-state models) m = 1,2



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



The Overall PDF

f (wi)=p(Si=m) f (wi |Si=m)

p(Si=m) . . . PMF of the hidden states∑M

m=1 p(Si=m) = 1

f (wi |Si = m) . . . conditional probability of the coefficientsvalue wi given the state Si corresponds to G(µi,m, σ

2i,m)

Transition ProbabilitiesChildren hidden states Si given the parent state Sp(i)

f (Si =m | Sp(i) =n)=

[f (Si =1 | Sp(i) =1) f (Si =1 | Sp(i) =2)f (Si =2 | Sp(i) =1) f (Si =2 | Sp(i) =2)

]For M =2 (2-state model)Persistence⇒ f1,1>> f2,1, f2,2>> f1,2



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



The Overall PDF

f (wi)=p(Si=m) f (wi |Si=m)

p(Si=m) . . . PMF of the hidden states∑M

m=1 p(Si=m) = 1

f (wi |Si = m) . . . conditional probability of the coefficientsvalue wi given the state Si corresponds to G(µi,m, σ

2i,m)

Transition ProbabilitiesChildren hidden states Si given the parent state Sp(i)

f (Si =m | Sp(i) =n)=

[f (Si =1 | Sp(i) =1) f (Si =1 | Sp(i) =2)f (Si =2 | Sp(i) =1) f (Si =2 | Sp(i) =2)

]For M =2 (2-state model)Persistence⇒ f1,1>> f2,1, f2,2>> f1,2



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography



HMT Training

Tying within subbands to prevent model overfitting⇒3 independent HMT treesExpectation Maximization (EM) training algorithm



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Noise Reduction

Noise Reduction via HMTs

NoisyImage

WaveletAnalysis

Alteringwavelet

coefficients

Via HMT models

WaveletSynthesis


DenoisedImage



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Noise Reduction

Additional Noise ModelWavelet domain, a noisy w. coefficient observation wi

wi = yi + ni

y . . . desired noise-free signal, n . . . iid Gaussian noise

Noise ReductionConditional mean estimate of yi , given the observedcoefficients w and the HMT model parameters θ

E [yi |w, θ] =M∑

m=1

p(Si = m) ·σ2

i,m

σ2n + σ2

i,m· wi (1)

p(Si =m), σi,m . . . obtained from the HMT modelσn . . . noise std. deviation (MAD estimate)



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Noise Reduction

Additional Noise ModelWavelet domain, a noisy w. coefficient observation wi

wi = yi + ni

y . . . desired noise-free signal, n . . . iid Gaussian noise

Noise ReductionConditional mean estimate of yi , given the observedcoefficients w and the HMT model parameters θ

E [yi |w, θ] =M∑

m=1

p(Si = m) ·σ2

i,m

σ2n + σ2

i,m· wi (1)

p(Si =m), σi,m . . . obtained from the HMT modelσn . . . noise std. deviation (MAD estimate)



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Noise Reduction

Altering Wavelet Coefficients

Noise reduction via HMT and NormalShrink



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Results

Residual Images Parameters in Our Experiments

Noise NormalShrink HMT

µn [10−2] σ2n [10−2] µ [10−2] σ2 [10−2] µ [10−2] σ2 [10−2]

5.00 3.00 0.04 2.18 1.12 0.600.00 1.00 0.00 1.12 0.16 0.325.00 1.00 0.46 1.04 1.00 0.32

Residual image - difference between the noisereduction result and the original image



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Results

Noise Reduction Results (a) ORIGINAL

(f) ABS. DIFFERENCE NORMALSHRINK (e) ABS. DIFFERENCE HMT (d) NOISY IMAGE

(c) DENOISED VIA NORMALSHRINK (b) DENOISED VIA HMT



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Conclusions

Noise Reduction ExperimentsHMT models outperform the NormalShrink method(at the expense of greater computation cost)NormalShrink causes artifacts and blurs edges

Future WorkOur experiment have been very limited so farNext step: Carry out more experiments onbiomedical image data



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Conclusions

Noise Reduction ExperimentsHMT models outperform the NormalShrink method(at the expense of greater computation cost)NormalShrink causes artifacts and blurs edges

Future WorkOur experiment have been very limited so farNext step: Carry out more experiments onbiomedical image data



ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Table of Contents

1 Introduction



4 Results

5 Conclusions




ICT Prague

Introduction


NormalShrink


Wavelet-Based HMTs

Noise Reduction

Results

Conclusions

Bibliography


Further Reading

M. S. Crouse, R. D. Nowak, and R. G. Baraniuk.Wavelet-Based Statistical Signal Processing Using Hidden MarkovModels.IEEE Trans. on Signal Processing, 46(4):886–902, April, 1998.

H. Choi and R. G. Baraniuk.Multiscale Image Segmentation Using Wavelet Domain HiddenMarkov Models.Int. Conf. on Image Processing, 1309–1321, IEEE, 2001.

C. W. Shaffrey, N. G. Kingsbury, and I. H. Jermyn.Unsupervised Image Segmentation via Markov Trees and ComplexWavelets.Int. Conf. on Image Processing, USA, 801–804, IEEE, 2002.

D. B. Percival and A. T. Walden.Wavelet Methods for Time Series Analysis.Cambridge University Press, USA, 2006.

L. Kaur, S. Gupta and R. C. ChauhanImage Denoising Using Wavelet Thresholding.3rd Conf. on Computer Vision, India, 1 – 4, 2002.

Wavelet Shrinkage FOR IMAGE NOISE REDUCTIONdsp.vscht.cz/hostalke/upload/TCP08_presentation.pdf · WAVELET HMT FOR NOISE REDUCTION ICT Prague Introduction Wavelet Shrinkage WS Scheme

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